The mechanic won´t change up to the rank of an admiral.
1: You gain a new rank.
2: You speak to the admiral on the earth space station.
3: You visit the upper level of the shipyard.
4: You get the the first vessel of your new tier for 1$.
5: If you want, you can buy further ones for energy credits, if you dont, thats it.
I'll preface this with: Some folks that do not know the correct answers, should endeavor to keep their posts to themselves. It only spreads misinformation and ignorance.
I am sure that this has been covered already, but just in case:
1. You get one ship per Rank for free.
2. You must spend a certain amount of skill points to level to the next Rank. That means you might have to go and spend a few points on a Lt. level skill Watch the two bars at the top (one white and a smaller one gold). You will gain the quest immediately upon allocating those points that were needed. Admiral Quinn will then give you your reward that will allow you to obtain your new rank's ship.
3. Getting the ship completes your rank-up.
4. Attending the promotion ceremony IS optional and can be dropped (or completed next time you are at Sol.)
I don't think many people here actually read the original poster's comments.
His rank is listed as Commander 1. That means he's spent enough skill points, because if he didn't, he'd be listed as Lieutenant Commander 11. Also, he's said he's spent skill points on commander skills. That can't happen until you've been promoted by spending enough skill pionts.. So, he's been promoted.
Typically, when this happens, you are immediately contacted (ie a comm window pops up immediately) by Admiral Quinn and told to return to starbase for promotion. If that didn't happen then it sounds like a bug.
I would check to see if you can get a promotion quest from Quinn or it it's in your mission log already. If not, submit a support ticket.
Make sure you spend your skill points too. Just advancing on the level bar isn't enough. The levels also have a skill point spent requirement. Unfortunately, I'm only a Lt. Cmdr right now, so can't tell you how many you need to spend for Commander.
Oh and if you hover your cursor over the bar itself, a little pop-up will tell you where you are and what you need.